## Fourier transformation can improve quadrature efficiency of Laplace distribution

### by Jinhyo Kim and Sangwon Seo .

Abstract: A numerical quadrature of a particular probability integral is concerned with using the Fourier transformation which smoothes the stiffness. \ The Fourier transformation of the Laplace distribution becomes, in a statistical sense, the Cauchy distribution. \ It is shown that the Gauss-Hermite quadrature of the Cauchy distribution, equivalent to the {\em Fourier-transformed} Laplace distribution, exhibits {\em better numerical} efficiency than the Gauss-Hermite quadrature of the {\em untransformed} Laplace distribution. \ A numerical example supports the analytic argument.

Key Words: Fourier transformation, Gauss-Hermite quadrature, Laplace distribution, Cauchy distribution

Authors:
Jinhyo Kim , jinkim@stats.snu.ac.kr
Sangwon Seo, sangwon@math.snu.ac.kr

Editor: John P. Hinde , J.P.Hinde@exeter.ac.uk

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